Temporal Abstractions and a Partitioning Heuristic for Interactive Resource Allocation

ed data: Atomic data: (Task (A B C)) (Task A) (Task B) (Task C) (Task (D E)) (Task D) (Task (F G)) (Task F) (Task G) (Task E) Figure 13: Abstract tasks resulting from summarization. 6 Value-Assignment Delay Heuristic This section proposes a new heuristic called the valueassignment delay heuristic (VAD) useful for solving list-coloring problems. An arc in the constraint graph of a resource allocation problem, expressed as a CSP as de ned in Section 2, indicates a mutually exclusive binary constraint between the nodes it links. This means that nodes linked cannot be given the same value. Two nodes are linked if and only if their time duration intersect and they have at least one value (i.e. resource) in common. Since in practical applications, many tasks are to be executed at the same time and many resources are solicited by many tasks at the same time, the corresponding constraint graphs tend to be densely connected. One can limit the high degree of interdependency among tasks by caching the shared resources. This operation, when applied repeatedly to the most commonly shared values, will result in a progressive elimination of arcs in the constraint graph, gradually separating it into independent clusters. 6.1 An illustrative example { a b } { a b } { a b c d } { a e } N1 N2 N3 N4 Figure 14: An example of the list-coloring problem. In Figure 14 we show a simple example of a list-coloring problem with four nodes: N1; N2; N3 and N4. Each node has a set of values it can be assigned. These sets are given in Figure 14. Links between two nodes indicate that they cannot be given the same value. A backtrack-search procedure that starts by assigning value a to either N1 or N2 will not nd a solution before going through a signi cant amount of backtracking. { a b } { a b } { a b c d } { a e } Initial Cluster N1 N2 N3 N4 Delay value {a} Cluster 2 { b } { b } { b c d } delay={a} delay={a} delay={a} { e } delay={a} N 2 Cluster 1 Delay value {b} N4 N1 N3 { } delay={a,b} Cluster 4 N4 { c d } delay={a,b} N1 Cluster 3 { } delay={a,b} Cluster 5 N3 Figure 15: Applying the value-assignment delay heuristic to a simple example.